Theorem sibfima 34797

Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)

Hypotheses

Ref	Expression
sitgval.b	⊢ 𝐵 = (Base‘𝑊)
sitgval.j	⊢ 𝐽 = (TopOpen‘𝑊)
sitgval.s	⊢ 𝑆 = (sigaGen‘𝐽)
sitgval.0	⊢ 0 = (0g‘𝑊)
sitgval.x	⊢ · = ( ·𝑠 ‘𝑊)
sitgval.h	⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitgval.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
sibfmbl.1	⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))

Assertion

Ref	Expression
sibfima	⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))

Proof of Theorem sibfima

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sibfmbl.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
2		sitgval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
3		sitgval.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊)
4		sitgval.s	. . . . 5 ⊢ 𝑆 = (sigaGen‘𝐽)
5		sitgval.0	. . . . 5 ⊢ 0 = (0g‘𝑊)
6		sitgval.x	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
7		sitgval.h	. . . . 5 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
8		sitgval.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
9		sitgval.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10	2, 3, 4, 5, 6, 7, 8, 9	issibf 34792	. . . 4 ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
11	1, 10	mpbid 235	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
12	11	simp3d 1162	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
13		sneq 4601	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
14	13	imaeq2d 6064	. . . . 5 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝐴}))
15	14	fveq2d 6889	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑀‘(◡𝐹 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝐴})))
16	15	eleq1d 2850	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)))
17	16	rspcv 3579	. 2 ⊢ (𝐴 ∈ (ran 𝐹 ∖ { 0 }) → (∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)))
18	12, 17	mpan9 516	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∖ cdif 3903  {csn 4591  ∪ cuni 4874  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666  ‘cfv 6540  (class class class)co 7419  Fincfn 8949  0cc0 11119  +∞cpnf 11259  [,)cico 13394  Basecbs 17295  Scalarcsca 17339   ·_𝑠 cvsca 17340  TopOpenctopn 17500  0_gc0g 17518  ℝHomcrrh 34451  sigaGencsigagen 34597  measurescmeas 34654  MblFnMcmbfm 34708  sitgcsitg 34788

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-sitg 34789

This theorem is used by:  sibfinima  34798  sitgfval  34800  sitgclg  34801